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CCSS.Math: ,

so we have an interesting shape right over here the base is a rectangular prism in the dimensions of that rectangular prism it's three units I guess we could say tall four units wide and then four units long and then on top of that on top of that we have what you could call a right pyramid where the the height of this right pyramid so if you start at the center of its base right over here and you go to the top this height right over here is one unit and this hasn't been drawn completely the scale and kind of the perspective skews it a little bit but our goal here our goal here is to figure out what is the length what is the length of one of these of one of these edges right over here so either that one or this one right over here what is that length and we will call that we will call that X and so I encourage you to pause this video and try to think about it on your own remember this is a right this is a right pyramid so what that tells us is is that this red line that is that's a one unit long it is perpendicular it is perpendicular to this entire plane it's perpendicular to the top of the rectangular of the rectangular prism so with that in mind I encourage you to pause the video and see if you can figure it out and I will give you a hint you will have to use the Pythagorean theorem maybe more than once alright so I assuming you you've at least given it a shot so let's work through it together so the key here is to realize well okay this this point this base right over you this point right over here it's half way in this direction half way in this direction so we can figure out well this entire this entire length right over here is length four so halfway this is going to be I'll write it with perspective that's going to be two and that's going to be two just like that and then the other thing we can figure out we can figure out what this this length is going to be because once again it's half way in that direction so if this whole thing is to and we see it right over here this is a rectangular prism so this length is going to be the same thing as this length so this whole thing is to then each of these this is going to be one and this is going to be one right over there well how does that help us well using that information we should be able to figure out we should be able to figure out this length see let me which I'll keep it in this color - this color is easy to see we should be able to figure out this length well why is this length interesting well if we know that length that length forms a right triangle that length and the one are the two non hypotenuse sides of a right triangle and then the X would be the hypotenuse so we could just apply the Pythagorean theorem so if we can figure out this we can figure out X so let's do one step at a time how do we figure out how do we figure out on oh let me call this length how do we call that how do we figure out length a well let's just take it out and look at it in two dimensions so if we look at it in two dimensions if we look at it in two dimensions it would look something like this so that's our length a we know that this length is half of this side right over here so that's going to be one and actually let me do it the same colors so this right over here is the same thing as this right over here and it is going to be of length one and then this right over here is going to be the same as this right over here which is going to be of length two and so we can just use the Pythagorean theorem here we know that the hypotenuse squared is going to be equal to 1 squared 1 squared plus 2 squared 1 squared plus 2 squared which of course is equal to 5 sorry is equal to this 1 plus 4 which is equal to 5 so we could write a let me do this in the magenta color we could write a squared is equal to 5 or we could say that a is equal to the principal root of 5 so this length right over here is the square root of five the principle root of five and now we can use that information to solve for X and let's take this right triangle and it sticks a little bit of visualization practice to visualize right but notice this is a right triangle this this height right here of length one is perpendicular to this entire plane so let's see if I can draw so this so we have this side is the square root of five and then we have a height let me do it looks like it's in a well let's say we're talking about this height right over here this height right over here is length one so I'll draw that that is of length one and we are trying to figure out X we are trying to figure out X in orange so we're trying to figure out we're trying to figure out this X and once again we know this is a right triangle this is a right triangle so we can apply the Pythagorean theorem again so we will have x squared is equal to one squared which is just I'll just write it 1 squared plus square root of five squared plus square root of five squared well that gets us x squared is equal to one plus five right square root of five squared is just five so one plus five is equal to six so we get X is equal to the square root of six and we're done we figured out the length of this side right over here